# Basics of internal forced convection

(Difference between revisions)
 Revision as of 01:07, 5 June 2010 (view source)← Older edit Revision as of 01:10, 5 June 2010 (view source)Newer edit → Line 29: Line 29: $\frac{L_{H}}{D}\ge 0.05\operatorname{Re}\text{ for laminar flow}$ $\frac{L_{H}}{D}\ge 0.05\operatorname{Re}\text{ for laminar flow}$ - | {{EquationRef|(1)}} + | {{EquationRef|(2)}} |} |} {| class="wikitable" border="0" {| class="wikitable" border="0" Line 36: Line 36: $\frac{L_{H}}{D}\ge 0.625\operatorname{Re}^{0.25}\text{ for turbulent flow}$ $\frac{L_{H}}{D}\ge 0.625\operatorname{Re}^{0.25}\text{ for turbulent flow}$ - | {{EquationRef| (1)}} + | {{EquationRef| (3)}} |} |} where LH is the hydrodynamic length and the Reynolds number is defined by$\operatorname{Re}=\frac{u_{m}D}{\nu }$ where LH is the hydrodynamic length and the Reynolds number is defined by$\operatorname{Re}=\frac{u_{m}D}{\nu }$ Line 61: Line 61: $\frac{u}{u_{c}}\quad \ \text{or}\quad \ \frac{u}{u_{m}}=f\left( \frac{r}{r_{o}} \right)$ $\frac{u}{u_{c}}\quad \ \text{or}\quad \ \frac{u}{u_{m}}=f\left( \frac{r}{r_{o}} \right)$ - | {{EquationRef| (1)}} + | {{EquationRef| (4)}} |} |} Fully developed temperature profile Fully developed temperature profile Line 69: Line 69: $\theta =\frac{T_{w}-T}{T_{w}-T_{m}}\quad \ \text{or}\quad \ \frac{T_{w}-T}{T_{w}-T_{c}}=g\left( \frac{r}{r_{o}} \right)$ $\theta =\frac{T_{w}-T}{T_{w}-T_{m}}\quad \ \text{or}\quad \ \frac{T_{w}-T}{T_{w}-T_{c}}=g\left( \frac{r}{r_{o}} \right)$ - | {{EquationRef| (1)}} + | {{EquationRef| (5)}} |} |} Fully developed concentration profile Fully developed concentration profile Line 77: Line 77: $\varphi =\frac{c_{w}-c}{c_{w}-c_{m}}\quad \ \text{or}\quad \ \frac{c_{w}-c}{c_{w}-c_{c}}=h\left( \frac{r}{r_{o}} \right)$ $\varphi =\frac{c_{w}-c}{c_{w}-c_{m}}\quad \ \text{or}\quad \ \frac{c_{w}-c}{c_{w}-c_{c}}=h\left( \frac{r}{r_{o}} \right)$ - | {{EquationRef| (1)}} + | {{EquationRef| (6)}} |} |} We can now define the local heat and mass transfer coefficients (h and hm) based on the mean temperature or concentration. We can now define the local heat and mass transfer coefficients (h and hm) based on the mean temperature or concentration. Line 85: Line 85: ${q}''_{w}=h\left( T_{w}-T_{m} \right)=-\left. k\frac{\partial T}{\partial r} \right|_{r=r_{o}}$ ${q}''_{w}=h\left( T_{w}-T_{m} \right)=-\left. k\frac{\partial T}{\partial r} \right|_{r=r_{o}}$ - | {{EquationRef| (1)}} + | {{EquationRef| (7)}} |} |} {| class="wikitable" border="0" {| class="wikitable" border="0" Line 92: Line 92: $\dot{{m}''}_{w}=h_{m}\left( \omega _{w}-\omega _{m} \right)=\left. -\rho D\frac{\partial \omega }{\partial r} \right|_{r=r_{o}}$ $\dot{{m}''}_{w}=h_{m}\left( \omega _{w}-\omega _{m} \right)=\left. -\rho D\frac{\partial \omega }{\partial r} \right|_{r=r_{o}}$ - | {{EquationRef| (1)}} + | {{EquationRef| (8)}} |} |} where D is mass diffusivity. where D is mass diffusivity. Line 101: Line 101: $\left. \frac{\partial }{\partial r}\left( \frac{T_{w}-T}{T_{w}-T_{m}} \right) \right|_{r=r_{o}}=\text{constant}=\frac{\left. -\frac{\partial T}{\partial r} \right|_{r=r_{o}}}{T_{w}-T_{m}}=\frac{h}{k}=\text{constant}$ $\left. \frac{\partial }{\partial r}\left( \frac{T_{w}-T}{T_{w}-T_{m}} \right) \right|_{r=r_{o}}=\text{constant}=\frac{\left. -\frac{\partial T}{\partial r} \right|_{r=r_{o}}}{T_{w}-T_{m}}=\frac{h}{k}=\text{constant}$ - | {{EquationRef| (1)}} + | {{EquationRef| (9)}} |} |} The above conclusion, that the local heat transfer coefficient is constant along the flow direction for a fully developed temperature profile, is only valid for constant wall heat flux or constant wall temperature conditions. The above conclusion, that the local heat transfer coefficient is constant along the flow direction for a fully developed temperature profile, is only valid for constant wall heat flux or constant wall temperature conditions. Line 110: Line 110: $\frac{\partial }{\partial x}\left( \frac{T_{w}-T}{T_{w}-T_{m}} \right)=0$ $\frac{\partial }{\partial x}\left( \frac{T_{w}-T}{T_{w}-T_{m}} \right)=0$ - | {{EquationRef| (1)}} + | {{EquationRef| (10)}} |} |} Differentiating the above equation yields Differentiating the above equation yields Line 118: Line 118: $\frac{\partial T}{\partial x}=\frac{dT_{w}}{dx}-\left( \frac{T_{w}-T}{T_{w}-T_{m}} \right)\frac{dT_{w}}{dx}+\left( \frac{T_{w}-T}{T_{w}-T_{m}} \right)\frac{dT_{m}}{dx}$ $\frac{\partial T}{\partial x}=\frac{dT_{w}}{dx}-\left( \frac{T_{w}-T}{T_{w}-T_{m}} \right)\frac{dT_{w}}{dx}+\left( \frac{T_{w}-T}{T_{w}-T_{m}} \right)\frac{dT_{m}}{dx}$ - | {{EquationRef| (1)}} + | {{EquationRef| (11)}} |} |} In external flow, the heat and mass transfer coefficients are usually defined by a driving differential (Tw - T∞) or (ωw – ω∞) where T∞ and ω∞ are the temperature and mass fraction of the fluid in the free stream (far away from the wall). In most cases, T∞ and ω∞ are known and constant for external flows. However, in internal flow configurations, there is not usually a well-defined temperature or concentration (mass fraction), except at the inlet and/or the boundaries. In internal flow, the temperature and concentrations may change both in the axial direction and perpendicular to the flow direction. Therefore, there are several choices available for the driving differential for temperature and concentration in internal flow. In external flow, the heat and mass transfer coefficients are usually defined by a driving differential (Tw - T∞) or (ωw – ω∞) where T∞ and ω∞ are the temperature and mass fraction of the fluid in the free stream (far away from the wall). In most cases, T∞ and ω∞ are known and constant for external flows. However, in internal flow configurations, there is not usually a well-defined temperature or concentration (mass fraction), except at the inlet and/or the boundaries. In internal flow, the temperature and concentrations may change both in the axial direction and perpendicular to the flow direction. Therefore, there are several choices available for the driving differential for temperature and concentration in internal flow. Line 127: Line 127: $T_{m}=\frac{1}{Au_{m}\rho _{m}c_{p,m}}\int_{A}^{{}}{uT\rho c_{p}}dA$ $T_{m}=\frac{1}{Au_{m}\rho _{m}c_{p,m}}\int_{A}^{{}}{uT\rho c_{p}}dA$ - | {{EquationRef| (1)}} + | {{EquationRef| (12)}} |} |} {| class="wikitable" border="0" {| class="wikitable" border="0" Line 134: Line 134: $\rho _{A,m}=\frac{1}{Au_{m}}\int_{A}^{{}}{u\rho _{A}}dA$ $\rho _{A,m}=\frac{1}{Au_{m}}\int_{A}^{{}}{u\rho _{A}}dA$ - | {{EquationRef| (1)}} + | {{EquationRef| (13)}} |} |} where $\rho _{A,m}$ is the mean mass density for a given component A, and $\rho _{m}$ is the mean density for the fluid where the mean velocity is defined as where $\rho _{A,m}$ is the mean mass density for a given component A, and $\rho _{m}$ is the mean density for the fluid where the mean velocity is defined as Line 142: Line 142: $u_{m}=\frac{1}{A\rho _{m}}\int_{A}^{{}}{u\rho dA}$ $u_{m}=\frac{1}{A\rho _{m}}\int_{A}^{{}}{u\rho dA}$ - | {{EquationRef| (1)}} + | {{EquationRef| (14)}} |} |} Assuming constant properties for mean velocity, temperature and mass density in the above equation, we obtain Assuming constant properties for mean velocity, temperature and mass density in the above equation, we obtain Line 150: Line 150: $u_{m}=\frac{1}{A}\int_{A}^{{}}{udA}$ $u_{m}=\frac{1}{A}\int_{A}^{{}}{udA}$ - | {{EquationRef| (1)}} + | {{EquationRef| (15)}} |} |} {| class="wikitable" border="0" {| class="wikitable" border="0" Line 157: Line 157: $T_{m}=\frac{1}{Au_{m}}\int_{A}^{{}}{uTdA}$ $T_{m}=\frac{1}{Au_{m}}\int_{A}^{{}}{uTdA}$ - | {{EquationRef| (1)}} + | {{EquationRef| (16)}} |} |} {| class="wikitable" border="0" {| class="wikitable" border="0" Line 164: Line 164: $\rho _{A,m}=\frac{1}{Au_{m}}\int_{A}^{{}}{\rho _{A}udA}$ $\rho _{A,m}=\frac{1}{Au_{m}}\int_{A}^{{}}{\rho _{A}udA}$ - | {{EquationRef| (1)}} + | {{EquationRef| (17)}} |} |} We will now focus our attention on two special conventional boundary conditions; constant wall heat flux and constant surface temperature. First, consider the constant heat flux or heat rate at the wall, which occurs in many applications such as electronic cooling, electric resistance heating, and radiant heating. From eq. (5.9), since h and ${q}''_{w}$ are constant, we can conclude Tw – Tm = constant. Differentiating leads to We will now focus our attention on two special conventional boundary conditions; constant wall heat flux and constant surface temperature. First, consider the constant heat flux or heat rate at the wall, which occurs in many applications such as electronic cooling, electric resistance heating, and radiant heating. From eq. (5.9), since h and ${q}''_{w}$ are constant, we can conclude Tw – Tm = constant. Differentiating leads to Line 177: Line 177: $\frac{\partial T}{\partial x}=\frac{dT_{w}}{dx}=\frac{dT_{m}}{dx}$ $\frac{\partial T}{\partial x}=\frac{dT_{w}}{dx}=\frac{dT_{m}}{dx}$ - |{{EquationRef|(1)}} + |{{EquationRef|(18)}} |} |} Now consider the case of constant surface or wall temperature, which also occurs in many applications including condensers, evaporators and any heat exchange surface where the heat transfer coefficient is extremely high. Using eq. (5.13) and the fact that dTw /dx = 0 for constant surface temperature, we get Now consider the case of constant surface or wall temperature, which also occurs in many applications including condensers, evaporators and any heat exchange surface where the heat transfer coefficient is extremely high. Using eq. (5.13) and the fact that dTw /dx = 0 for constant surface temperature, we get Line 186: Line 186: $\frac{\partial T}{\partial x}=\left( \frac{T_{w}-T}{T_{w}-T_{m}} \right)\frac{dT_{m}}{dx}$ $\frac{\partial T}{\partial x}=\left( \frac{T_{w}-T}{T_{w}-T_{m}} \right)\frac{dT_{m}}{dx}$ - |{{EquationRef|(1)}} + |{{EquationRef|(19)}} |} |} [[Image:Fig5.4.png|thumb|400 px|alt=Wall and mean temperature variation along the flow in a circular tube for fully developed flow and temperature profile |Figure 4: Wall and mean temperature variation along the flow in a circular tube for fully developed flow and temperature profile.]] [[Image:Fig5.4.png|thumb|400 px|alt=Wall and mean temperature variation along the flow in a circular tube for fully developed flow and temperature profile |Figure 4: Wall and mean temperature variation along the flow in a circular tube for fully developed flow and temperature profile.]] Line 203: Line 203: $\frac{dp}{dx}=\frac{\mu }{r}\frac{d}{dr}\left( r\frac{du}{dr} \right)$ $\frac{dp}{dx}=\frac{\mu }{r}\frac{d}{dr}\left( r\frac{du}{dr} \right)$ - |{{EquationRef|(1)}} + |{{EquationRef|(20)}} |} |} {| class="wikitable" border="0" {| class="wikitable" border="0" Line 221: Line 221: \end{align}[/itex] \end{align}[/itex] - | {{EquationRef|(1)}} + | {{EquationRef|(21)}} |} |} Integrating the above equation twice and using the boundary conditions yield a parabolic velocity profile: Integrating the above equation twice and using the boundary conditions yield a parabolic velocity profile: Line 230: Line 230: $u=\frac{-r_{o}^{2}}{4\mu }\left( \frac{dp}{dx} \right)\left( 1-\frac{r^{2}}{r_{o}^{2}} \right)$ $u=\frac{-r_{o}^{2}}{4\mu }\left( \frac{dp}{dx} \right)\left( 1-\frac{r^{2}}{r_{o}^{2}} \right)$ - |{{EquationRef|(1)}} + |{{EquationRef|(22)}} |} |} Using the definition of mean velocity, Using the definition of mean velocity, Line 241: Line 241: $u_{m}=\frac{\int_{A}^{{}}{udA}}{A}=\frac{\int_{0}^{\text{ }r_{o}}{2\pi rudr}}{\pi r_{o}^{2}}=-\frac{r_{o}^{2}}{8\mu }\frac{dp}{dx}$ $u_{m}=\frac{\int_{A}^{{}}{udA}}{A}=\frac{\int_{0}^{\text{ }r_{o}}{2\pi rudr}}{\pi r_{o}^{2}}=-\frac{r_{o}^{2}}{8\mu }\frac{dp}{dx}$ - |{{EquationRef|(1)}} + |{{EquationRef|(23)}} |} |} Equation (5.24) in terms of mean velocity is Equation (5.24) in terms of mean velocity is Line 250: Line 250: $u=2u_{m}\left( 1-\frac{r^{2}}{r_{o}^{2}} \right)$ $u=2u_{m}\left( 1-\frac{r^{2}}{r_{o}^{2}} \right)$ - |{{EquationRef|(1)}} + |{{EquationRef|(24)}} |} |} The shear stress at the wall can be calculated from the velocity gradient at the wall. The shear stress at the wall can be calculated from the velocity gradient at the wall. Line 259: Line 259: $\tau _{w}=\left. \mu \frac{\partial u}{\partial r} \right|_{r=r_{o}}=\frac{4u_{m}\mu }{r_{o}}=-\frac{r_{o}}{2}\frac{dp}{dx}$ $\tau _{w}=\left. \mu \frac{\partial u}{\partial r} \right|_{r=r_{o}}=\frac{4u_{m}\mu }{r_{o}}=-\frac{r_{o}}{2}\frac{dp}{dx}$ - |{{EquationRef|(1)}} + |{{EquationRef|(25)}} |} |} The above result can be presented in terms of the friction coefficient, cf. The above result can be presented in terms of the friction coefficient, cf. Line 268: Line 268: $c_{f}=\frac{\tau _{w}}{\rho u_{m}^{2}/2}=\frac{8\mu }{r_{o}\rho u_{m}}=\frac{16}{\operatorname{Re}}$ $c_{f}=\frac{\tau _{w}}{\rho u_{m}^{2}/2}=\frac{8\mu }{r_{o}\rho u_{m}}=\frac{16}{\operatorname{Re}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(26)}} |} |} In addition to the above friction coefficient, the following friction factor is also widely used: In addition to the above friction coefficient, the following friction factor is also widely used: Line 277: Line 277: $f=\frac{-(dp/dx)D}{\rho u_{m}^{2}/2}$ $f=\frac{-(dp/dx)D}{\rho u_{m}^{2}/2}$ - |{{EquationRef|(1)}} + |{{EquationRef|(27)}} |} |} It follows from eq. (5.27) that It follows from eq. (5.27) that Line 286: Line 286: $f=\frac{4\tau _{w}}{\rho u_{m}^{2}/2}=4c_{f}=\frac{64}{\operatorname{Re}}$ $f=\frac{4\tau _{w}}{\rho u_{m}^{2}/2}=4c_{f}=\frac{64}{\operatorname{Re}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(28)}} |} |} Let’s make an analysis of the energy equation to get a feeling about the importance of various terms, since determination of the temperature field in the fluid is required for the heat transfer coefficients. To simplify the analysis, lets consider a two-dimensional cylindrical geometry with the following assumptions: Let’s make an analysis of the energy equation to get a feeling about the importance of various terms, since determination of the temperature field in the fluid is required for the heat transfer coefficients. To simplify the analysis, lets consider a two-dimensional cylindrical geometry with the following assumptions: Line 300: Line 300: $u\frac{\partial T}{\partial x}=\alpha \left[ \frac{1}{r}\frac{\partial \left( r\partial T/\partial r \right)}{\partial r}+\frac{\partial ^{2}T}{\partial x^{2}} \right]+\frac{\mu }{\rho c_{p}}\left( \frac{\partial u}{\partial r} \right)^{2}$ $u\frac{\partial T}{\partial x}=\alpha \left[ \frac{1}{r}\frac{\partial \left( r\partial T/\partial r \right)}{\partial r}+\frac{\partial ^{2}T}{\partial x^{2}} \right]+\frac{\mu }{\rho c_{p}}\left( \frac{\partial u}{\partial r} \right)^{2}$ - |{{EquationRef|(1)}} + |{{EquationRef|(29)}} |} |} The above equation is non-dimensionalized using the following variables to show the effect of axial conduction and viscous dissipation: The above equation is non-dimensionalized using the following variables to show the effect of axial conduction and viscous dissipation: Line 311: Line 311: \end{matrix}[/itex] \end{matrix}[/itex] - | {{EquationRef|(1)}} + | {{EquationRef|(30)}} |} |} where Tr is a reference temperature and E and Pe are Eckert and Peclet numbers, respectively. where Tr is a reference temperature and E and Pe are Eckert and Peclet numbers, respectively. Line 321: Line 321: $\frac{u^{+}}{2}\frac{\partial \theta }{\partial x^{+}}=\frac{1}{r^{+}}\frac{\partial }{\partial r^{+}}\left( r^{+}\frac{\partial \theta }{\partial r^{+}} \right)+\frac{1}{2Pe^{2}}\frac{\partial ^{2}\theta }{\partial x^{+2}}+E\Pr \left( \frac{\partial u^{+}}{\partial r^{+}} \right)^{2}$ $\frac{u^{+}}{2}\frac{\partial \theta }{\partial x^{+}}=\frac{1}{r^{+}}\frac{\partial }{\partial r^{+}}\left( r^{+}\frac{\partial \theta }{\partial r^{+}} \right)+\frac{1}{2Pe^{2}}\frac{\partial ^{2}\theta }{\partial x^{+2}}+E\Pr \left( \frac{\partial u^{+}}{\partial r^{+}} \right)^{2}$ - |{{EquationRef|(1)}} + |{{EquationRef|(31)}} |} |} The second term on the right hand side of the above equation is due to axial heat conduction, and the last term is due to the viscous dissipation effect. If E Pr is small, viscous dissipation can be neglected. This is true for flow with a low velocity and low Prandtl number. The second term on the right hand side (axial heat conduction) is neglected when the Peclet number, Pe, is greater than 100. Axial heat conduction should be accounted for when the Peclet number is small, in the case of liquid metals. The second term on the right hand side of the above equation is due to axial heat conduction, and the last term is due to the viscous dissipation effect. If E Pr is small, viscous dissipation can be neglected. This is true for flow with a low velocity and low Prandtl number. The second term on the right hand side (axial heat conduction) is neglected when the Peclet number, Pe, is greater than 100. Axial heat conduction should be accounted for when the Peclet number is small, in the case of liquid metals.

## Revision as of 01:10, 5 June 2010

It is important to clarify some basic definitions, terminologies and criteria that are often used in internal convective heat and mass transfer. These include:
1. Mean velocity, temperature, and concentration
2. Fully developed flow, temperature, and concentration profiles
3. Hydrodynamic, thermal, and concentration entrance lengths

Figure 1(a) shows the development of a velocity profile inside a duct or tube with uniform inlet velocity for laminar flow of an incompressible Newtonian fluid. The velocity profile at some distance away from the tube’s inlet no longer changes along the flow direction, where it is referred to as the fully developed flow condition. The fully developed condition is often met at some distance away from the inlet. However, there are also applications in which fully developed flow is never reached. Momentum, thermal, and concentration boundary layers form on the inside surface of the tube. The thickness of the layers increases in a similar manner as boundary layer flow over a flat plate (which was presented in detail in Chapter 4). Figure 1(a) shows how the momentum boundary layer builds up in a pipe along the flow direction. At some distance away from the inlet, the boundary layer fills the flow area. The flow downstream from this point is referred to as fully developed flow since the velocity slope does not change after this point. The distance downstream from the inlet to where the flow becomes fully developed is called the hydrodynamic entrance length. If the flow is laminar (Re < 2300 for flow inside circular tubes), the fully developed velocity is a parabolic shape. It should be noted that the fluid velocity outside the boundary layer increases with x, which is required to satisfy the conservation of mass (or continuity) equation.

The center line velocity finally reaches a value two times the inlet velocity, uin, for fully developed, steady, incompressible, laminar flow inside tubes. It should be noted that the hydrodynamic entrance length for fully developed flow does not start from the point where the friction coefficient, $c_{f}=\frac{\tau _{w}}{\rho u_{in}^{2}/2}$ (1)

does not change along the flow. The friction coefficient variation for laminar flow inside a circular tube with uniform inlet velocity is shown in Fig. 1(b). The friction coefficient is highest at the entrance and then decreases smoothly to a constant value, corresponding to fully developed flow. Two factors cause the friction coefficient to be higher in the entrance region of tubes than in the fully developed region. The first factor is the larger velocity gradient at the entrance on the wall. The gradient decreases along the pipe and becomes constant before the velocity becomes fully developed. The second factor is the velocity outside the boundary layer, which must increase to satisfy the conservation of mass or continuity equation. Accelerating velocity in the core produces an additional drag force when its effect is considered in the friction coefficient.

The turbulent velocity profile and friction coefficient variation for a circular pipe are shown in Fig. 2. Even for a very high inlet velocity, the boundary layer will be laminar over a part of the entrance. This transition from laminar to turbulent is clearly shown by the sudden increase in momentum boundary layer thickness as shown in Fig. 2(a). The friction coefficient variation for turbulent flow in a pipe entrance is shown in Fig. 2(b). The hydrodynamic entry length required for fully developed flow should be obtained by a complete solution of the flow and thermal field in the entrance region. A rule of thumb to judge whether or not the flow is fully developed for circular pipes is $\frac{L_{H}}{D}\ge 0.05\operatorname{Re}\text{ for laminar flow}$ (2) $\frac{L_{H}}{D}\ge 0.625\operatorname{Re}^{0.25}\text{ for turbulent flow}$ (3)

where LH is the hydrodynamic length and the Reynolds number is defined by $\operatorname{Re}=\frac{u_{m}D}{\nu }$

A similar behavior is expected for thermal cases with the thermal boundary layer growth at the entrance of a tube as shown in Figure 3, which corresponds to a case where there may be an unheated length in which the velocity is fully developed before heating starts. One expects that the thermal boundary layer increases in the thermal entry region before the heat transfer coefficient becomes constant. It should be noted that the requirement for a fully developed thermal region is that the dimensionless temperature, θ, $\theta =\frac{T_{w}-T}{T_{w}-T_{m}}\quad \ \text{or}\quad \ \frac{T_{w}-T}{T_{w}-T_{c}}$

does not change with distance along the flow direction. The mean and centerline temperatures are Tm and Tc, respectively. Similar requirements exist for the fully developed concentration profile where θ is replaced with $\varphi$, $\varphi =\frac{c_{w}-c}{c_{w}-c_{m}}\quad \ \text{or}\quad \ \frac{c_{w}-c}{c_{w}-c_{c}}$

where the mean and centerline concentrations (or mass fractions) are cm and cc, respectively. In the subsequent sections, we use the following definitions to mathematically define the fully developed flow, temperature, and concentration profiles: Fully developed flow profile $\frac{u}{u_{c}}\quad \ \text{or}\quad \ \frac{u}{u_{m}}=f\left( \frac{r}{r_{o}} \right)$ (4)

Fully developed temperature profile $\theta =\frac{T_{w}-T}{T_{w}-T_{m}}\quad \ \text{or}\quad \ \frac{T_{w}-T}{T_{w}-T_{c}}=g\left( \frac{r}{r_{o}} \right)$ (5)

Fully developed concentration profile $\varphi =\frac{c_{w}-c}{c_{w}-c_{m}}\quad \ \text{or}\quad \ \frac{c_{w}-c}{c_{w}-c_{c}}=h\left( \frac{r}{r_{o}} \right)$ (6)

We can now define the local heat and mass transfer coefficients (h and hm) based on the mean temperature or concentration. ${q}''_{w}=h\left( T_{w}-T_{m} \right)=-\left. k\frac{\partial T}{\partial r} \right|_{r=r_{o}}$ (7) $\dot{{m}''}_{w}=h_{m}\left( \omega _{w}-\omega _{m} \right)=\left. -\rho D\frac{\partial \omega }{\partial r} \right|_{r=r_{o}}$ (8)

where D is mass diffusivity. Since we define the fully developed temperature profile as when the non-dimensional temperature profile (TwT) / (TwTm) is invariant in the flow direction (x-direction), we can write the following equation: $\left. \frac{\partial }{\partial r}\left( \frac{T_{w}-T}{T_{w}-T_{m}} \right) \right|_{r=r_{o}}=\text{constant}=\frac{\left. -\frac{\partial T}{\partial r} \right|_{r=r_{o}}}{T_{w}-T_{m}}=\frac{h}{k}=\text{constant}$ (9)

The above conclusion, that the local heat transfer coefficient is constant along the flow direction for a fully developed temperature profile, is only valid for constant wall heat flux or constant wall temperature conditions. The requirement for the dimensionless temperature to be invariant for a fully developed temperature profile can also be presented as $\frac{\partial }{\partial x}\left( \frac{T_{w}-T}{T_{w}-T_{m}} \right)=0$ (10)

Differentiating the above equation yields $\frac{\partial T}{\partial x}=\frac{dT_{w}}{dx}-\left( \frac{T_{w}-T}{T_{w}-T_{m}} \right)\frac{dT_{w}}{dx}+\left( \frac{T_{w}-T}{T_{w}-T_{m}} \right)\frac{dT_{m}}{dx}$ (11)

In external flow, the heat and mass transfer coefficients are usually defined by a driving differential (Tw - T∞) or (ωw – ω∞) where T∞ and ω∞ are the temperature and mass fraction of the fluid in the free stream (far away from the wall). In most cases, T∞ and ω∞ are known and constant for external flows. However, in internal flow configurations, there is not usually a well-defined temperature or concentration (mass fraction), except at the inlet and/or the boundaries. In internal flow, the temperature and concentrations may change both in the axial direction and perpendicular to the flow direction. Therefore, there are several choices available for the driving differential for temperature and concentration in internal flow. The most common choice for defining the driving temperature or concentration is based on mean temperature or concentration (mass fraction or mass density). The mixed mean fluid temperature or concentration is defined at a given local axial location based on the convective thermal energy or mass balance, i.e., $T_{m}=\frac{1}{Au_{m}\rho _{m}c_{p,m}}\int_{A}^{{}}{uT\rho c_{p}}dA$ (12) $\rho _{A,m}=\frac{1}{Au_{m}}\int_{A}^{{}}{u\rho _{A}}dA$ (13)

where ρA,m is the mean mass density for a given component A, and ρm is the mean density for the fluid where the mean velocity is defined as $u_{m}=\frac{1}{A\rho _{m}}\int_{A}^{{}}{u\rho dA}$ (14)

Assuming constant properties for mean velocity, temperature and mass density in the above equation, we obtain $u_{m}=\frac{1}{A}\int_{A}^{{}}{udA}$ (15) $T_{m}=\frac{1}{Au_{m}}\int_{A}^{{}}{uTdA}$ (16) $\rho _{A,m}=\frac{1}{Au_{m}}\int_{A}^{{}}{\rho _{A}udA}$ (17)

We will now focus our attention on two special conventional boundary conditions; constant wall heat flux and constant surface temperature. First, consider the constant heat flux or heat rate at the wall, which occurs in many applications such as electronic cooling, electric resistance heating, and radiant heating. From eq. (5.9), since h and q''w are constant, we can conclude Tw – Tm = constant. Differentiating leads to $\frac{dT_{w}}{dx}=\frac{dT_{m}}{dx}$

Substituting into eq. (5.13) gives us $\frac{\partial T}{\partial x}=\frac{dT_{w}}{dx}=\frac{dT_{m}}{dx}$ (18)

Now consider the case of constant surface or wall temperature, which also occurs in many applications including condensers, evaporators and any heat exchange surface where the heat transfer coefficient is extremely high. Using eq. (5.13) and the fact that dTw /dx = 0 for constant surface temperature, we get $\frac{\partial T}{\partial x}=\left( \frac{T_{w}-T}{T_{w}-T_{m}} \right)\frac{dT_{m}}{dx}$ (19) Figure 4: Wall and mean temperature variation along the flow in a circular tube for fully developed flow and temperature profile.

It should be emphasized that eqs. (5.20) and (5.21) are only applicable when the temperature profile is fully developed. The variations of wall and mean temperature for the fully developed temperature profile along the flow for constant heat rate or surface temperature are shown in Fig. 4. Finally, to obtain the convective heat and/or mass transfer coefficients, one needs to solve the continuity, mass, momentum, energy and appropriate species equations. In convective heat and mass transfer problems, it is important to obtain information about the flow by solving the continuity and momentum equations, in addition to the energy and species equations. These conservation equations are mostly decoupled, except for circumstances such as a variable property, or coupled governing equations or boundary conditions due to physical circumstances (which happens in applications such as natural convection, absorption, sublimation, evaporation and condensation problems).

It is obviously more accurate to solve the complete transport conservation equations (elliptic form) for internal flow without making boundary layer assumptions (parabolic form) as discussed in Chapter 4. However, in most cases it is not practical due to complexity of the geometry and/or solution techniquesm, as well as the requirement of additional boundary conditions in both analytical or numerical methods. For the case of two-dimensional fully developed steady laminar flow with constant properties, the momentum equation in a circular tube, including boundary conditions, as shown in Chapter 2 is: $\frac{dp}{dx}=\frac{\mu }{r}\frac{d}{dr}\left( r\frac{du}{dr} \right)$ (20) \begin{align} & u=0\begin{matrix} {} & {} \\ \end{matrix}\text{at}\begin{matrix} {} & {} \\ \end{matrix}r=r_{\text{o}} \\ & \frac{du}{dr}=0\begin{matrix} {} & {} \\ \end{matrix}\text{at}\begin{matrix} {} & {} \\ \end{matrix}r=0 \\ \end{align} (21)

Integrating the above equation twice and using the boundary conditions yield a parabolic velocity profile: $u=\frac{-r_{o}^{2}}{4\mu }\left( \frac{dp}{dx} \right)\left( 1-\frac{r^{2}}{r_{o}^{2}} \right)$ (22)

Using the definition of mean velocity, um , for constant properties and the above equation, we obtain $u_{m}=\frac{\int_{A}^{{}}{udA}}{A}=\frac{\int_{0}^{\text{ }r_{o}}{2\pi rudr}}{\pi r_{o}^{2}}=-\frac{r_{o}^{2}}{8\mu }\frac{dp}{dx}$ (23)

Equation (5.24) in terms of mean velocity is $u=2u_{m}\left( 1-\frac{r^{2}}{r_{o}^{2}} \right)$ (24)

The shear stress at the wall can be calculated from the velocity gradient at the wall. $\tau _{w}=\left. \mu \frac{\partial u}{\partial r} \right|_{r=r_{o}}=\frac{4u_{m}\mu }{r_{o}}=-\frac{r_{o}}{2}\frac{dp}{dx}$ (25)

The above result can be presented in terms of the friction coefficient, cf. $c_{f}=\frac{\tau _{w}}{\rho u_{m}^{2}/2}=\frac{8\mu }{r_{o}\rho u_{m}}=\frac{16}{\operatorname{Re}}$ (26)

In addition to the above friction coefficient, the following friction factor is also widely used: $f=\frac{-(dp/dx)D}{\rho u_{m}^{2}/2}$ (27)

It follows from eq. (5.27) that $f=\frac{4\tau _{w}}{\rho u_{m}^{2}/2}=4c_{f}=\frac{64}{\operatorname{Re}}$ (28)

Let’s make an analysis of the energy equation to get a feeling about the importance of various terms, since determination of the temperature field in the fluid is required for the heat transfer coefficients. To simplify the analysis, lets consider a two-dimensional cylindrical geometry with the following assumptions: 1. Steady laminar flow 2. Constant properties 3. Fully developed flow 4. Newtonian incompressible fluid The energy equation under the above assumptions is $u\frac{\partial T}{\partial x}=\alpha \left[ \frac{1}{r}\frac{\partial \left( r\partial T/\partial r \right)}{\partial r}+\frac{\partial ^{2}T}{\partial x^{2}} \right]+\frac{\mu }{\rho c_{p}}\left( \frac{\partial u}{\partial r} \right)^{2}$ (29)

The above equation is non-dimensionalized using the following variables to show the effect of axial conduction and viscous dissipation: $\begin{matrix} u^{+}=\frac{u}{u_{m}}\quad ,\quad x^{+}=\frac{2\left( x/D \right)}{\operatorname{Re}\Pr }\quad ,\quad \text{E}=\frac{u_{m}^{2}}{c_{p}\Delta T} \\ \theta =\frac{T-T_{r}}{T_{in}-T_{r}}\quad ,\quad \Delta T=T_{in}-T_{r}\quad ,\quad r^{+}=\frac{r}{r_{o}}\quad ,\quad \text{Pe}=\operatorname{Re}\Pr \\ \end{matrix}$ (30)

where Tr is a reference temperature and E and Pe are Eckert and Peclet numbers, respectively. The resulting dimensionless energy equation is $\frac{u^{+}}{2}\frac{\partial \theta }{\partial x^{+}}=\frac{1}{r^{+}}\frac{\partial }{\partial r^{+}}\left( r^{+}\frac{\partial \theta }{\partial r^{+}} \right)+\frac{1}{2Pe^{2}}\frac{\partial ^{2}\theta }{\partial x^{+2}}+E\Pr \left( \frac{\partial u^{+}}{\partial r^{+}} \right)^{2}$ (31)

The second term on the right hand side of the above equation is due to axial heat conduction, and the last term is due to the viscous dissipation effect. If E Pr is small, viscous dissipation can be neglected. This is true for flow with a low velocity and low Prandtl number. The second term on the right hand side (axial heat conduction) is neglected when the Peclet number, Pe, is greater than 100. Axial heat conduction should be accounted for when the Peclet number is small, in the case of liquid metals.